Answer:
Let's differentiate each of the given expressions step by step:
(i) Differentiating e^(m.cos^(-1)(x)) * sin(nx) with respect to x:
To differentiate the product of two functions, we can use the product rule:
(d/dx) [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)
Here, f(x) = e^(m.cos^(-1)(x)) and g(x) = sin(nx).
To find the derivative of f(x), we can use the chain rule. Let's define h(x) = m.cos^(-1)(x):
f(x) = e^(h(x))
f'(x) = e^(h(x)) * h'(x)
To find the derivative of h(x), we can use the chain rule again. Let's define k(x) = cos^(-1)(x):
h(x) = m * k(x)
h'(x) = m * k'(x)
To find the derivative of k(x), we differentiate the inverse cosine function:
k(x) = cos^(-1)(x)
k'(x) = -1 / sqrt(1 - x^2)
Now, let's substitute the derivatives into the product rule formula:
= [e^(m.cos^(-1)(x)) * m * (-1 / sqrt(1 - x^2))] * sin(nx) + e^(m.cos^(-1)(x)) * sin(nx) * n * cos(nx)
= -m * e^(m.cos^(-1)(x)) * sin(nx) / sqrt(1 - x^2) + n * e^(m.cos^(-1)(x)) * sin(nx) * cos(nx)
(ii) To differentiate the expression (e^x^2) * tan^(-1)(x) / sqrt(1 + x^2), we will use the product rule and the chain rule.
Let's break down the expression into separate parts:
Part 1: Differentiating e^x^2
To differentiate e^x^2, we can use the chain rule:
(d/dx) [e^(x^2)] = e^(x^2) * 2x
Part 2: Differentiating tan^(-1)(x)
To differentiate tan^(-1)(x), we can use the chain rule:
(d/dx) [tan^(-1)(x)] = 1 / (1 + x^2)
Part 3: Differentiating sqrt(1 + x^2)
To differentiate sqrt(1 + x^2), we can use the chain rule:
(d/dx) [sqrt(1 + x^2)] = (1/2) * (1 + x^2)^(-1/2) * 2x
Now, let's apply the product rule to the expression:
(d/dx) [(e^x^2) * tan^(-1)(x) / sqrt(1 + x^2)]
= (e^x^2) * 1 / (1 + x^2) * (1/2) * (1 + x^2)^(-1/2) * 2x + (e^x^2) * tan^(-1)(x) * (1/2) * (1 + x^2)^(-1/2) * 2x - (e^x^2) * tan^(-1)(x) * (1/2) * (1 + x^2)^(-3/2) * 2x
Simplifying further:
= e^x^2 * x / (1 + x^2) * (1 + x^2)^(-1/2) + e^x^2 * tan^(-1)(x) * (1 + x^2)^(-1/2) * 2x - e^x^2 * tan^(-1)(x) * (1 + x^2)^(-3/2) * 2x
= e^x^2 * x / (1 + x^2) * (1 + x^2)^(-1/2) + 2x * e^x^2 * tan^(-1)(x) * (1 + x^2)^(-1/2) - 2x * e^x^2 * tan^(-1)(x) * (1 + x^2)^(-3/2)
Simplifying the exponents:
= e^x^2 * x / (1 + x^2) + 2x * e^x^2 * tan^(-1)(x) / sqrt(1 + x^2) - 2x * e^x^2 * tan^(-1)(x) / (1 + x^2) * sqrt(1 + x^2)
Therefore, the derivative of (e^x^2) * tan^(-1)(x) / sqrt(1 + x^2) with respect to x is:
e^x^2 * x / (1 + x^2) + 2x * e^x^2 * tan^(-1)(x) / sqrt(1 + x^2) - 2x * e^x^2 * tan^(-1)(x) / (1 + x^2) * sqrt(1 + x^2)
(iii) Differentiating √x (2x + 3)^2 / √(x + 1) with respect to x:
To differentiate this expression, we can use the quotient rule:
If we have a function f(x) = g(x) / h(x), then the derivative of f(x) is given by:
f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2
Here, g(x) = √x (2x + 3)^2 and h(x) = √(x + 1).
Let's find the derivatives of g(x) and h(x):
g'(x) = d/dx [√x (2x + 3)^2]
= (1/2) * (2x + 3)^2 * d/dx [√x] + √x * d/dx [(2x + 3)^2]
= (1/2) * (2x + 3)^2 * (1 / (2√x)) + √x * 2(2x + 3)
= (2x + 3)^2 / (2√x) + 4√x (2x + 3)
h'(x) = d/dx [√(x + 1)]
= (1/2) * (x + 1)^(-1/2) * d/dx [(x + 1)]
= (1/2) * (x + 1)^(-1/2)
Now, let's substitute the derivatives into the quotient rule formula:
= [(2x + 3)^2 / (2√x) + 4√x (2x + 3)] * √(x + 1) - √x (2x + 3)^2 * (1/2) * (x + 1)^(-1/2) / [√(x + 1)]^2
= [(2x + 3)^2 / (2√x) + 4√x (2x + 3)] * √(x + 1) - √x (2x + 3)^2 * (1/2) * (x + 1)^(-1/2) / (x + 1)
= [(2x + 3)^2 * √(x + 1)] / (2√x) + 4√x (2x + 3)√(x + 1) - √x (2x + 3)^2 / (2√(x + 1))
(iv) Differentiating √((x - 3) (x^2 + 4) /(3x^2 + 4x + 5)) with respect to x:
To differentiate this expression, we can use the quotient rule and the chain rule.
Let's define the numerator as g(x) = √((x - 3) (x^2 + 4)) and the denominator as h(x) = √(3x^2 + 4x + 5).
To find the derivative of g(x), we can use the chain rule:
g'(x) = (1/2) * √((x - 3) (x^2 + 4)) * [(x - 3)' (x
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Verified answer
Answer:
Let's differentiate each of the given expressions step by step:
(i) Differentiating e^(m.cos^(-1)(x)) * sin(nx) with respect to x:
To differentiate the product of two functions, we can use the product rule:
(d/dx) [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)
Here, f(x) = e^(m.cos^(-1)(x)) and g(x) = sin(nx).
To find the derivative of f(x), we can use the chain rule. Let's define h(x) = m.cos^(-1)(x):
f(x) = e^(h(x))
f'(x) = e^(h(x)) * h'(x)
To find the derivative of h(x), we can use the chain rule again. Let's define k(x) = cos^(-1)(x):
h(x) = m * k(x)
h'(x) = m * k'(x)
To find the derivative of k(x), we differentiate the inverse cosine function:
k(x) = cos^(-1)(x)
k'(x) = -1 / sqrt(1 - x^2)
Now, let's substitute the derivatives into the product rule formula:
(d/dx) [f(x) * g(x)] = f'(x) * g(x) + f(x) * g'(x)
= [e^(m.cos^(-1)(x)) * m * (-1 / sqrt(1 - x^2))] * sin(nx) + e^(m.cos^(-1)(x)) * sin(nx) * n * cos(nx)
= -m * e^(m.cos^(-1)(x)) * sin(nx) / sqrt(1 - x^2) + n * e^(m.cos^(-1)(x)) * sin(nx) * cos(nx)
(ii) To differentiate the expression (e^x^2) * tan^(-1)(x) / sqrt(1 + x^2), we will use the product rule and the chain rule.
Let's break down the expression into separate parts:
Part 1: Differentiating e^x^2
To differentiate e^x^2, we can use the chain rule:
(d/dx) [e^(x^2)] = e^(x^2) * 2x
Part 2: Differentiating tan^(-1)(x)
To differentiate tan^(-1)(x), we can use the chain rule:
(d/dx) [tan^(-1)(x)] = 1 / (1 + x^2)
Part 3: Differentiating sqrt(1 + x^2)
To differentiate sqrt(1 + x^2), we can use the chain rule:
(d/dx) [sqrt(1 + x^2)] = (1/2) * (1 + x^2)^(-1/2) * 2x
Now, let's apply the product rule to the expression:
(d/dx) [(e^x^2) * tan^(-1)(x) / sqrt(1 + x^2)]
= (e^x^2) * 1 / (1 + x^2) * (1/2) * (1 + x^2)^(-1/2) * 2x + (e^x^2) * tan^(-1)(x) * (1/2) * (1 + x^2)^(-1/2) * 2x - (e^x^2) * tan^(-1)(x) * (1/2) * (1 + x^2)^(-3/2) * 2x
Simplifying further:
= e^x^2 * x / (1 + x^2) * (1 + x^2)^(-1/2) + e^x^2 * tan^(-1)(x) * (1 + x^2)^(-1/2) * 2x - e^x^2 * tan^(-1)(x) * (1 + x^2)^(-3/2) * 2x
= e^x^2 * x / (1 + x^2) * (1 + x^2)^(-1/2) + 2x * e^x^2 * tan^(-1)(x) * (1 + x^2)^(-1/2) - 2x * e^x^2 * tan^(-1)(x) * (1 + x^2)^(-3/2)
Simplifying the exponents:
= e^x^2 * x / (1 + x^2) + 2x * e^x^2 * tan^(-1)(x) / sqrt(1 + x^2) - 2x * e^x^2 * tan^(-1)(x) / (1 + x^2) * sqrt(1 + x^2)
Therefore, the derivative of (e^x^2) * tan^(-1)(x) / sqrt(1 + x^2) with respect to x is:
e^x^2 * x / (1 + x^2) + 2x * e^x^2 * tan^(-1)(x) / sqrt(1 + x^2) - 2x * e^x^2 * tan^(-1)(x) / (1 + x^2) * sqrt(1 + x^2)
(iii) Differentiating √x (2x + 3)^2 / √(x + 1) with respect to x:
To differentiate this expression, we can use the quotient rule:
If we have a function f(x) = g(x) / h(x), then the derivative of f(x) is given by:
f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2
Here, g(x) = √x (2x + 3)^2 and h(x) = √(x + 1).
Let's find the derivatives of g(x) and h(x):
g'(x) = d/dx [√x (2x + 3)^2]
= (1/2) * (2x + 3)^2 * d/dx [√x] + √x * d/dx [(2x + 3)^2]
= (1/2) * (2x + 3)^2 * (1 / (2√x)) + √x * 2(2x + 3)
= (2x + 3)^2 / (2√x) + 4√x (2x + 3)
h'(x) = d/dx [√(x + 1)]
= (1/2) * (x + 1)^(-1/2) * d/dx [(x + 1)]
= (1/2) * (x + 1)^(-1/2)
Now, let's substitute the derivatives into the quotient rule formula:
f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2
= [(2x + 3)^2 / (2√x) + 4√x (2x + 3)] * √(x + 1) - √x (2x + 3)^2 * (1/2) * (x + 1)^(-1/2) / [√(x + 1)]^2
= [(2x + 3)^2 / (2√x) + 4√x (2x + 3)] * √(x + 1) - √x (2x + 3)^2 * (1/2) * (x + 1)^(-1/2) / (x + 1)
= [(2x + 3)^2 * √(x + 1)] / (2√x) + 4√x (2x + 3)√(x + 1) - √x (2x + 3)^2 / (2√(x + 1))
(iv) Differentiating √((x - 3) (x^2 + 4) /(3x^2 + 4x + 5)) with respect to x:
To differentiate this expression, we can use the quotient rule and the chain rule.
Let's define the numerator as g(x) = √((x - 3) (x^2 + 4)) and the denominator as h(x) = √(3x^2 + 4x + 5).
To find the derivative of g(x), we can use the chain rule:
g'(x) = (1/2) * √((x - 3) (x^2 + 4)) * [(x - 3)' (x